I think it might be a good idea to backup at this point and recapitulate the physical theory we are developing, based on the two fundamental assumptions of Larson’s new physical system, the Reciprocal System of Physical Theory (RST) . We start off with a unit progression of space and time that we can represent mathematically with the rational numbers of the Reciprocal System of Mathematics (RSM). The RSM is based on an operational interpretation of rational numbers, and recognizes that the difference between the operational and quantitative interpretation of a rational number enables us to interpret a fraction, not as a part of a whole number, but as a whole number, and these whole numbers have two “directions” naturally; that is, they form a system of integers, where the negative integers are the inverse of the positive integers, represented as rational numbers, operationally interpreted.
On this basis, 1/2 is the inverse of 2/1, where 1/2 has a value of -1 and 2/1 has a value of +1, relative to 1/1. Of course, we can still reinterpret these numbers, quantitatively, when the need arises, by changing the reference datum of the value proposition from 1/1 to 0, where zero is the absolute value of nothing. In other words, quantitatively interpreted, the value of 1/2 lies between 0 and 1, and this interpretation is appropriately expressed as a decimal number, .5.
The significance in taking the RSM approach to numbers is not to be understood in terms of advocating a change in the traditional way of using numbers in physics, but in opening up the possibility for investigating a new way of thinking about physical magnitudes. In the RST, we assume that the entire structure of the physical universe is a structure of space and time; that is, the observed universe is not assumed to be a structure of space and time coordinates in which matter, radiation, and energy are contained and interact, via mysterious forces of interaction, causing objects to move, generating momentum and heat. Instead, all constituents of the universe are regarded as forms of space and time, or motion, and there is no container.
The starting point for this universe of motion is a unit expansion of space and time that forms the unit datum of magnitudes in the universe. This datum is dynamic, because it constitutes a ratio of change and, while it is represented by the number 1/1, it is a ratio of two rates of change, space/time = ds/dt.= 1/1. However, to indicate the new operational interpretation of this ratio, as contrasted with the traditional quantitative interpretation, we drop the division symbol, “/”, and replace it with the pipe symbol “|” that indicates that this is not a numerical quotient, but a reciprocal relation between the expansion of space and the expansion of time.
Consequently, this approach is tantamount to redefining space as the reciprocal of time in the equation of motion and assuming that the entire structure of the physical universe proceeds from this definition. Of course, this is not possible unless there is some possibility for a physical change in the unit ratio to occur, just as we would have no other numbers than 1|1, if both the numerator and denominator could only be changed together: The progression 1|1, 2|2. 3|3, …n|n is a uniform and unchanging unit progression. The only way to derive anything interesting from this progression is to introduce a change of some sort.
For instance, If we change the numerator in such a way that it increases twice as much as the denominator, with each change, so that the progression is no longer a unit change, but is now 1|1, 4|2, 6|3, …2n|n, then we have changed the rate of progression to ds|dt = 2|1, which is a positive unit of change in the progression rate. Similarly, to form a negative unit rate in the progression, we need only to change the denominator twice as much as the numerator, with each change, so that the progression is changed from 1|1 to 1|2, in the other “direction,” 1|1, 2|4, 3|6, …n|2n, which is a negative unit change in the progression rate, ds/dt = 1/2.
Physically, the only way to affect this change in the unit progression rate is to assume that the “direction” of the progression of one aspect of the motion changes by one unit each two units of progression, while the reciprocal aspect continues to increase uniformly. Thus, to get a change from the unit rate of progression, we assume that one aspect of the progression reverses its “direction,” from increasing, to decreasing, and back to increasing, continuously alternating the “direction” of its progression, which in effect cuts in half its rate of increase, relative to the rate of its reciprocal aspect’s increase.
For instance, if the space aspect of the progression begins this alternating pattern at a given location, it would increase, or expand for one unit, and decrease, or contract, for one unit, while the time aspect increased uniformly for two units, or twice as many units. Hence, the total units of progression is two units in each case, but only the time aspect has increased two units, while the space aspect has decreased and increased one unit, thereby changing the progression ratio of increase, from ds/dt = 1|1, to ds|dt = 1|2.
By the same token, if the time aspect of the progression begins the alternating pattern at a given location, the space|time ratio at that location becomes ds|dt = 2|1. These two, non-unit, rates of space|time progression are the fundamental units of the LRC’s RST-based, physical, theory. In this theory, the negative unit is designated the space unit-displacement ratio (SUDR), and the positive unit is designated the time unit-displacement ratio (TUDR).
Our theoretical development differs from Larson’s RST-based physical theory, in that the space|time “direction” reversals are treated differently. In our new theory, these fundamental units of motion are not rotated in various ways, as they are in Larson’s development, but are simply combined together in various combinations, in a way that is very similar to the way that Sundance Bilson-Thompson, the Australian physicist, has combined twists of space|time “ribbons,” in his topological toy model of preons, the name given to LST theories of sub-entities, forming the fermions of the standard model (SM).
In the Bilson-Thompson (BT) model, the magnitude of the twists are measured in terms of ± π radians. In this way, a “left” twist, rotating the ribbon - π radians, represents a magnitude of a half-cycle of rotation, while a “right” twist, rotating the ribbon + π radians, represents a magnitude of a half-cycle of rotation in the other “direction.” Therefore, combined together, two left twists represent a magnitude of a full-cycle magnitude of rotation in one “direction,” while two right twists represent a full-cycle magnitude in the other “direction.” Combining a left and right twist together creates an effective zero magnitude.
Lee Smolin and Fotini Markopoulou are now collaborating with BT in an attempt to take his ideas from a toy model of topography to a physical theory of spacetime quantum gravity with “local excitations that can be mapped to the first generation femions of the standard model of particle physics.” (see http://www.arxiv.org/ftp/hep-th/papers/0603/0603022.pdf)
However, the important thing for us is that the twists of spacetime “ribbons” in the BT model correspond to the space|time contraction/expansion, the SUDRs and TUDRs, in our theoretical model, where a half-twist corresponds to half of a contraction/expansion cycle of a SUDR or TUDR. In the BT model, the twists, referred to as “tweedles,” can be combined into what BT terms “helons.” He writes:
The three possible combinations of [tweedles] UU, EE, and UE = EU can be represented as ribbons bearing twists through the angles +2 π, −2 π, and 0 respectively. A twist through ±2 π is interpreted as an electric charge of ±e/3. We shall refer to such pairs of tweedles as helons (evoking their helical structure) and denote the three types of helons by H+, H−, and H0.
Thus, clearly, the topological magnitudes of helons in the BT-model correspond to the space|time magnitudes of the SUDR (H-), TUDR (H+), and the SUDR|TUDR combo (H0), in our theory. However, in the BT model, only Ho helons are combined with H+ and H- helons, and the respective magnitudes are magnitudes of M2 motion. The two positive and negative helons are never combined, because combining two instances of M2 motion, produces no motion. In our RST-based theory, on the other hand, the combination is a result of the mathematical relationship of the SUDRs and TUDRs, and, as we have shown below in the world line charts, the combination of a SUDR and TUDR produces the unit magnitude of (1|2)+(1|1)+(2|1) = 4|4, M4, motion, not the zero magnitude of (-1)+(1) = 0, M2, motion.
Accordingly, it is the combination of SUDRs and TUDRs into the S|T combos that produces the S|T entity, which is analogous to the H0 helon in the BT model, but differs in that it represents unit motion. Adding SUDRs and/or TUDRs to the fundamental S|T combo then creates the three types of S|T combos, as balanced in two “directions” (green), as unbalanced in the negative “direction” (red), and as unbalanced in the positive “direction” (blue).
Combining the three colors of S|Ts into triangular triplets then produces the first generation of fermions and bosons of the SM, as discussed in the previous posts below, following the patterns of helons in the BT model, developed with the help of group theory.
However, as far as I know, neither BT nor his collaborators have extended their work to the second and third generations of fermions as yet, but, as S|T combos are easily extended into higher dimensional objects, doing this in our “preon” theory, looks very promising.
Obviously, combining the entities of the S|T version of the SM, such as quarks and leptons into hadrons and atoms of the periodic table, so that they come out with the appropriate qualitative and quantitative properties of these subatoms and atoms, is the task of the LRC’s Microcosmic Research Division. Working with the consequences of these developments in the field of condensed matter physics is the task of the Macrocosmic Research Division, and working with the consequences with all of these in the field of cosmology is the task of the Macrocosmic Research Division, with all of these divisions depending upon the work of the LRC’s Mathematics Research Division.
Hence, regardless of whether or not a verifiably correct theory of both continuous and discrete physical magnitudes (so-called “theory of everything”) will eventually emerge from the work of the LRC, we can now clearly see that a new science has been born, an inductive science of physics that is testable and predictive, that is at once exceedingly simple and complex, and, while it is consonant with the LST science of the past, it is truly revolutionary in its concepts, systems, and results. The excitement is palpable.